RO-TST 2015 Problem 11

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .